What is the Talmudic and Halachic source for having this tiny flickering light in the Shul above the Aron HaKodesh, called a Ner Tamid, and what does it symbolize, other than having one more thing in Shul with a plaque honouring the donor or the dearly departed in whose memory.

The Ner Tamid, also known as the eternal light, usually Hangs above the Aron Hakodesh in every synagogue. It is a symbol of Hashem’s eternal and imminent presence in our congregations. Medresh Hagodol (Tetzave), Medresh Chazis (2), Trumas Hadeshen (345) Mekor Chaim (150; 9), Binyan Tziyon (12), Kochabei Yitzchok (12), Yerushas Pleito (6), and others mention that it is in memory of the Menorah in the Beis Hamikdosh. Sheloh (Torah Shebeksav) mentions that it is ‘merumaz’ or implied in the word ‘Tamid’ used in the Menorah of theTemple, signalling that it will last for all generations, even after the Churban Beis Hamikdash. Maharam Shik (Y.D. 374) relates about a Rov that on the inauguration of his shul when he lighted for the first time the Ner Tamid, recited complete brocho: ‘Lehadlik Ner Tamid.’ however, the author disagrees. Curiously, Mateh Aharon, presents a negative position regarding the Ner Tamid in a shul and maintains it seems to follow the ways of the Gentiles and should not be used.